Lévy scaling: the diffusion entropy analysis applied to DNA sequences.

Abstract

We address the problem of the statistical analysis of a time series generated by complex dynamics with the diffusion entropy analysis (DEA) [N. Scafetta, P. Hamilton, and P. Grigolini, Fractals 9, 193 (2001)]. This method is based on the evaluation of the Shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations, rather than on the measurement of the variance of this diffusion process, as done with the traditional methods. We compare the DEA to the traditional methods of scaling detection and prove that the DEA is the only method that always yields the correct scaling value, if the scaling condition applies. Furthermore, DEA detects the real scaling of a time series without requiring any form of detrending. We show that the joint use of DEA and variance method allows to assess whether a time series is characterized by Lévy or Gauss statistics. We apply the DEA to the study of DNA sequences and prove that their large-time scales are characterized by Lévy statistics, regardless of whether they are coding or noncoding sequences. We show that the DEA is a reliable technique and, at the same time, we use it to confirm the validity of the dynamic approach to the DNA sequences, proposed in earlier work.